Math has this weird way of looking terrifying when it’s actually just two small steps in a trench coat. You see $9^{3/2}$ on a test or in a coding algorithm and your brain probably does a little skip. It looks like it should be complicated. It isn't. Honestly, most people struggle with fractional exponents because they try to do everything at once. They see the 9, they see the 3, they see the 2, and it becomes a jumble of numbers.
The answer is 27.
That’s it. If you came here just for the number, there you go. But if you’re trying to understand why it’s 27, or why this specific math property shows up in everything from computer graphics to engineering, we need to talk about how powers and roots actually dance together. It’s less about calculation and more about a specific kind of logic.
Breaking Down 9 to the power of 3/2
When you look at $9^{3/2}$, you’re looking at a fractional exponent. Think of the fraction as a set of instructions. The top number (the numerator) is the power. The bottom number (the denominator) is the root. You can do them in any order you want.
Seriously.
You could square root the 9 first and then cube the result. Or you could cube the 9 and then take the square root. One of these is way easier than the other. If you cube 9 first, you're doing $9 \times 9 \times 9$, which is 729. Now you have to find the square root of 729. Unless you're a human calculator, that’s going to take a second.
But if you take the square root of 9 first? That’s 3. Then you cube it ($3 \times 3 \times 3$). Boom. 27. It's the same destination, just a much nicer walk.
This is the Power of a Power Rule in action. Formally, it looks like this:
$$a^{m/n} = (\sqrt[n]{a})^m = \sqrt[n]{a^m}$$
In our case, $a$ is 9, $m$ is 3, and $n$ is 2.
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Why the denominator is always the root
It helps to think about what a square root actually represents. A square root of 9 is just 9 to the power of 1/2. When you multiply $9^{1/2}$ by $9^{1/2}$, you add the exponents. $1/2 + 1/2 = 1$. So $9^1$ is 9. This is why the denominator represents the "root" part of the operation. It’s the divisor of the exponential growth.
Kinda cool, right?
Most people learn this in 8th or 9th grade and then immediately flush it out of their systems. But then you hit a college-level physics course or try to program a shader for a video game, and suddenly these fractional powers are everywhere. They are the backbone of non-linear scaling.
Real World Application: It’s Not Just Homework
You might wonder where $9^{3/2}$ actually lives outside of a textbook. Well, look at Kepler’s Third Law of Planetary Motion.
Johannes Kepler, a guy who spent way too much time looking at Mars, figured out that the square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit. Basically, if you want to find out how long it takes a planet to go around a star, you’re dealing with $a^{3/2}$.
If a planet is 9 units away from its sun, its orbital period is related to $9^{3/2}$.
It’s 27.
In the world of 3D rendering and technology, these types of calculations happen millions of times per second. When a light source hits a surface in a game like Cyberpunk 2077 or Minecraft, the falloff of that light—how it gets dimmer as it moves away—is often calculated using fractional exponents to create a "natural" look. Linear math looks fake. The world isn't linear. It’s exponential. It’s curvy.
Common Pitfalls to Avoid
I've seen people try to multiply the 9 by 3/2. That’s a massive mistake. $9 \times 1.5$ is 13.5. That is not how exponents work.
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An exponent is a repeated multiplication, but a fractional exponent is a hybrid of multiplication and division (roots). Another mistake is forgetting that the root must be a positive number when dealing with real numbers in this context. While you can take the cube root of a negative number, taking the square root of a negative number lands you in the "imaginary" territory of $i$.
Since 9 is positive, we don't have to worry about that here. We stay firmly in the realm of real numbers.
How to Calculate This on a Standard Calculator
Most people don't carry a scientific calculator anymore because your phone is one. But if you just type "9^3/2" into some basic calculators, they might mess up the order of operations. They might do $9^3$ and then divide the whole thing by 2.
If you're using a phone:
- Turn it sideways to get the scientific mode.
- Type 9.
- Hit the $x^y$ button.
- Type 1.5 (because 3 divided by 2 is 1.5).
- Hit equals.
You should see 27. If you see 364.5, your calculator did $9^3 = 729$ and then divided by 2. That’s wrong. It didn’t treat the 3/2 as a single exponent. You have to be smarter than the interface.
The Nuance of Radicals
Sometimes you'll see this written as $\sqrt{9^3}$ or $(\sqrt{9})^3$.
In professional mathematics, especially in calculus, we almost always prefer the fractional exponent notation ($9^{3/2}$) over the radical sign ($\sqrt{ }$). Why? Because it’s way easier to use the rules of derivatives and integrals when everything is in exponent form. If you’re a student heading toward STEM, get comfortable with the fraction. Stop looking for the "house" (the radical symbol). The fraction is your friend. It’s cleaner.
Visualizing the Growth
Imagine a square with a side length of 9. Its area is 81.
Now imagine a line of length 9.
The value of $9^{3/2}$ sits in this weird middle ground. It’s larger than the number itself (9) but smaller than the square (81).
When the exponent is between 1 and 2, you're looking at "super-linear" growth. It’s accelerating, but it hasn't quite hit the "explosive" stage of squaring.
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- $9^1 = 9$
- $9^{1.5} = 27$
- $9^2 = 81$
Notice the jump. Moving just 0.5 up in the exponent tripled the value (from 9 to 27). Moving another 0.5 tripled it again (from 27 to 81). This is the nature of exponential scales. They don't add; they multiply.
The Mental Shortcut Summary
Whenever you see a number to the 3/2 power, just remember: Root it, then Cube it.
- Look at the base (9).
- Take the square root ($\sqrt{9} = 3$).
- Cube the result ($3 \times 3 \times 3 = 27$).
If the base was 16, you’d do the same. Square root of 16 is 4. 4 cubed is 64. So $16^{3/2} = 64$.
If the base was 25? Square root is 5. 5 cubed is 125.
It becomes a game once you see the pattern. You stop fearing the fraction. You start seeing it as a shortcut. It's basically a "one-and-a-half" power.
Practical Steps for Mastery
If you want to actually get good at this so you never have to Google it again, try these three things:
Memorize your cubes up to 5.
Knowing that $2^3=8$, $3^3=27$, $4^3=64$, and $5^3=125$ makes these problems instant. You’ll see 27 and immediately think "3 cubed."
Practice the "Root First" rule.
Always try to shrink the number before you grow it. Cubing 9 is annoying. Square rooting 9 is easy. Always go for the smaller number first to keep the mental load low.
Apply it to different bases.
Try to calculate $4^{3/2}$ or $100^{3/2}$ in your head right now.
(For 4, it's $\sqrt{4}=2$, then $2^3=8$. For 100, it's $\sqrt{100}=10$, then $10^3=1000$).
Once you do this five times, the "fear" of the fraction disappears. You'll start noticing these patterns in financial compound interest formulas, in physics problems, and even in music theory frequencies. Math isn't about memorizing 27; it's about recognizing that 3/2 is just a specific way of growing a number that bridges the gap between a line and a square.